We introduce the concept of a pentagonal geometry as a generalization of the pentagon and the Desargues configuration, in the same vein that the generalized polygons share the fundamental properties of ordinary polygons. In short, a pentagonal geometry is a regular partial linear space in which for all points x, the points not collinear with the point x, form a line. We compute bounds on their parameters, give some constructions, obtain some nonexistence results for seemingly feasible parameters and suggest a cryptographic application related to identifying codes of partial linear spaces.

We elaborate on the existence and construction of the so-called combinatorial configurations. The main result is that for fixed degrees the existence of such configurations is given by a numerical semigroup. The proof is constructive giving a method to obtain combinatorial configurations with parameters large enough.

In this paper we discuss k-anonymous graphs in terms of modular decomposition and we present two algorithms for the k-anonymization of graphs with respect to neighborhoods. This is the strictest definition of k-anonymity for graphs. The first algorithm is an adaptation of the k-means algorithm to neighborhood clustering in graphs. The second algorithm is distributed of message passing type, and therefore enables user-privacy: the individuals behind the vertices can jointly protect their own privacy. Although these algorithms are not optimal in terms of information loss, they are a first example of algorithms that provide k-anonymization of graphs with respect to the strictest definition, and they are simple to implement.

We extend the list of known linear patterns admitted by numerical semigroups associated with combinatorial configurations. This is done through the construction of configurations from combinations of several smaller configurations. These results may be used to construct configurations with certain parameters, and therefore contribute with answers to the existence question for these configurations.

It is proved that a numerical semigroup can be associated to the triangle-free -configurations, and some results on existence are deduced. For example it is proved that for any there exists infinitely many -configurations. Most proofs are given from a graph theoretical point of view, in the sense that the configurations are represented by their incidence graphs. An application to private information retrieval is described.

It is proved that the numerical semigroups associated to the combinatorial configurations satisfy a family of non-linear symmetric patterns. Also, these numerical semigroups are studied for two particular classes of combinatorial configurations.

User-private information retrieval (UPIR) is the art of retrieving information without telling the information holder who you are. UPIR is sometimes called anonymous keyword search. This article discusses a UPIR protocol in which the users form a peer-to-peer network over which they collaborate in protecting the privacy of each other. The protocol is known as P2P UPIR. It will be explained why the P2P UPIR protocol may have a flaw in the protection of the privacy of the client in front of the server. Two alternative variations of the protocols are discussed. One of these will prove to resolve the privacy flaw discovered in the original protocol. Hence the aim of this article is to propose a modification of the P2P UPIR protocol. It is justified why the projective planes are still the optimal configurations for P2P UPIR for the modified protocol.

User-private information retrieval systems should protect the user’s anonymity when performing queries against a database, or they should limit the servers capacity of profiling users. Peer-to-peer user-private information retrieval (P2P UPIR) supplies a practical solution: the users in a group help each other in doing their queries, thereby preserving their privacy without any need of the database to cooperate. One way to implement the P2P UPIR uses combinatoric configurations to administrate the keys needed for the private communication between the peers.

This article is devoted to the choice of the configuration in this system. First of all we characterize the optimal configurations for the P2P UPIR and see the relationship with the projective planes as described in finite geometry. Then we give a very efficient construction of such optimal configurations, i.e. finite projective planes. We finally check that the involved graphs are Ramanujan graphs, giving an additional justification of the optimality of the constructed configurations.

A long list of personal tragedies, including teenage suicides, has raised the importance of managing the personal information available on the Internet. It has been argued that it should be allowed to make mistakes, and that there should be a right to be forgotten. Unfortunately, today's Internet architecture and services typically do not support such functionality. We design a system that provides digital oblivion for users of online social networks. Participants form a peer-based agent community, which agree on protecting the privacy of individuals who request images to be forgotten. The system distributes and maintains up-to-date information on oblivion requests, and implements a filtering functionality when accessing an underlying online social network. We describe digital oblivion in terms of authentication of user-to-content relations and identify two user-to-content relations that are particularly relevant for digital oblivion. Finally, we design a family of protocols that provide digital oblivion with respect to these user-to-content relations, within the community that are implementing the protocol. Our protocols leverage a combination of digital signatures, watermarking, image tags, and trust management. No collaboration is required from the social network provider, although the system could also be incorporated as a standard feature of the social network.

Anonymous database search protocols allow users to query a database anonymously. This can be achieved by letting the users form a peer-to-peer community and post queries on behalf of each other. In this article we discuss an application of combinatorial configurations (also known as regular and uniform partial linear spaces) to a protocol for anonymous database search, as defining the key-distribution within the user community that implements the protocol. The degree of anonymity that can be provided by the protocol is determined by properties of the neighborhoods and the closed neighborhoods of the points in the combinatorial configuration that is used. Combinatorial configurations with unique neighborhoods or unique closed neighborhoods are described and we show how to attack the protocol if such configurations are used. We apply k-anonymity arguments and present the combinatorial configurations with k-anonymous neighborhoods and with k-anonymous closed neighborhoods. The transversal designs and the linear spaces are presented as optimal configurations among the configurations with k-anonymous neighborhoods and k-anonymous closed neighborhoods, respectively.

In data privacy, the evaluation of the disclosure risk has to take into account the fact that several releases of the same or similar information about a population are common. In this paper we discuss this issue within the scope of k-anonymity. We also show how this issue is related to the publication of privacy protected databases that consist of linked tables. We present algorithms for the implementation of k-anonymity for this type of data.

In this paper we discuss some tools for graph perturbation with applications to data privacy. We present and analyse two different approaches. One is based on matrix decomposition and the other on graph partitioning. We discuss these methods and show that they belong to two traditions in data protection: noise addition/microaggregation and k-anonymity.

In this paper we discuss the relations between clustering and error correcting codes. We show that clustering can be used for constructing error correcting codes. We review the previous works found in the literature about this issue, and propose a modification of a previous work that can be used for code construction from a set of proposed codewords.

In this article we provide a formal framework for reidentification in general. We define n-confusion as a concept for modeling the anonymity of a database table and we prove that n-confusion is a generalization of k-anonymity. After a short survey on the different available definitions of k-anonymity for graphs we provide a new definition for k-anonymous graph, which we consider to be the correct definition. We provide a description of the k-anonymous graphs, both for the regular and the non-regular case. We also introduce the more flexible concept of (k, l)-anonymous graph. Our definition of (k, l)-anonymous graph is meant to replace a previous definition of (k, l)-anonymous graph, which we here prove to have severe weaknesses. Finally, we provide a set of algorithms for k-anonymization of graphs.

Fuzzy measures are monotonic set functions on a reference set; they generalize probabilities replacing the additivity condition by monotonicity. The typical application of these measures is with fuzzy integrals. Fuzzy integrals integrate a function with respect to a fuzzy measure, and they can be used to aggregate information from a set of sources (opinions from experts or criteria in a multicriteria decision-making problem). In this context, background knowledge on the sources is represented by means of the fuzzy measures. For example, interactions between criteria are represented by means of nonadditive measures. In this paper, we introduce fuzzy measures on multisets. We propose a general definition, and we then introduce a family of fuzzy measures for multisets which we show to be equivalent to distorted probabilities when the multisets are restricted to proper sets